40 _ MATHEMATICS, 
exhibited as a rectangle, until it touches the second cylinder ; the problem, 
therefore, is reduced to finding the part which intersects the surface of the 
first cylinder. In vertical projection this is easy, as here the line of inter- 
section coincides with the convexity of the cylinder; nothing more is 
necessary, then, than to prolong the corners @ and g to the cylinder EK. The 
case is different, however, in the horizontal projection, the line of intersection 
here being a curve. If the axes of the two cylinders lie in the same plane, 
the curve will be symmetrical; if, as in our example, this is not the case, 
the superior half will be different from the inferior, and it becomes necessary 
in all cases to seek similarly situated points of the intersection surface, in the 
horizontal and vertical projection. For this purpose, under ag and a’g’, 
describe the semicircles whose projections are the lines ag and a’g’; divide 
their circumferences into any number of equal parts, and draw lines parallel 
to the edges of the small cylinder. These must reach to the circumference 
of the larger cylinder, in the vertical projection, and may be of any length 
in the horizontal. From the points where these lines intersect the circum- 
ference of the larger cylinder in the vertical projection, let fall perpendiculars 
upon the corresponding parallels in the horizontal projection. We shall thus 
obtain the points h, 7, k, /, m,n, and 0, which are common to the convexities 
of both cylinders, and must consequently lie in the contour of the surface of 
intersection ; this latter may then be easily described. We have represented 
the lower half of the curve; the upper is obtained in a similar manner. As 
the cylinders approach towards equality in thickness, the curve becomes 
abrupt; when both are equal, the intersection appears in the projection as 
two straight lines, which meet above the axis of the cylinder. 
Pl. 4, fig. 33, represents the intersections of two cylinders, of different 
diameters, when their axes lie in the same plane. The mode of construct- 
ing the cylinders and their bases follows from what has already been said. 
We must remark, however, that the two upper ellipses in the vertical view 
have arisen from a misapprehension of the engraver ; the upper bases should 
have been projected as straight lines. The construction of the intersection 
follows from what was said in explanation of fig. 32. With respect to the 
horizontal projection, the views of the bases are readily found, these being 
ellipses, whose perpendicular axes are the respective diameters of the cylin- 
ders, the horizontal being determined from the vertical view, by means of 
the perpendiculars gg’, hh’, ee’, ff’, aa’, bb’, cc’, & dd’. To project the line 
of intersection, the points of division of the projection e’f’ are projected upon 
the ellipse ef, at 1’, 2’, &c., parallels to the surface, drawn through the points 
of the ellipse thus obtained, and at the corresponding points the line of 
intersection in the vertical view, cut by perpendiculars. The points of in- 
tersection will be common to both cylinders, or be points in the line of 
intersection. The lower line of intersection, dotted only in the figure, is 
obtained in a similar manner. 
Fig. 34 exhibits the intersection of a cylinder and a sphere, where the 
cylinder has the smaller diameter of the two, and is not parallel to the sur- 
face of projection. The development of both projections presents no difh- 
culty in itself, if what has already been said on the subject be kept in mind ; 
40 
